IS‐PINN: A Fast and Efficient Training of Physics‐Informed Neural Networks Using Invariant Subspace Reductions for Solving Nonlinear Diffusion Equations
Jiafu Pang, Shoufeng Shen, Guofu Yu, Junyang Zhang
Source abstract
ABSTRACT We propose the invariant subspace physics‐informed neural network (IS‐PINN), a novel approach that combines the invariant subspace method (ISM) with deep learning to solve nonlinear diffusion equations. By restricting the solution to a prescribed invariant subspace spanned by exact basis functions, IS‐PINN eliminates spatial dependencies and transforms the nonlinear diffusion equation into a dynamical system. The temporal coefficients are learned by a neural network with hard‐constrained initial conditions via least‐squares projection. We validate IS‐PINN on five problems: the semilinear heat equation, the Riabouchinsky–Proudman–Johnson equation (RPJE), the porous medium equation, the quasilinear parabolic equation, and the (1+2)D reaction–diffusion equation. Numerical results demonstrate that IS‐PINN outperforms standard PINNs, achieving mean relative errors of – over five random seeds, with mean training time reduced by approximately 74%–98%.
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